Development of a quadrature-based moment method for

Development of a quadrature-based moment method for polydisperse flows and incorporation in MFIXMFIX-QMOM implementation A verification test case
Development of a quadrature-based moment method for polydisperse flows and incorporation
in MFIX
Iowa State University - Department of Chemical and Biological Engineering
DOE Annual Review Meeting: Development, Verification and Validation of Multiphase Models for Polydisperse Flows
April 22th-23st, 2009 - Morgantown, WV.
A. Passalacqua, R. O. Fox Quadrature-based moment method for polydisperse flows
Overview Background on kinetic theory of gas-solid flow
MFIX-QMOM implementation A verification test case
Collocation in the project roadmap
Project roadmap
Collocation in the project roadmap
Targeted elements in the roadmap
A1-NT High-fidelity, transient, 3-D, two-phase with PSD (no density variations), hydrodynamics-only simulation of transport reactor.
B6-NT Identify the deficiencies of the current models, assess the state-of-the-art, and document the current best approach.
B8-NT Develop a plan for generating validation test cases, identify fundamental experiments, and identify computational challenge problems.
E3-NT Train adequate number of graduate students in this area.
Fundamental equations: Kinetic theory of granular flow
Particle phase kinetic equation
fi (vi, x, t): number density function of species i
vi: particle velocity
Fi: force acting on each particle (drag, gravity, . . . )
Cij: rate of change of fi due to collisions with species j
Fluid phase equations of motion
∂
βf,i: Drag coefficient of species i.
Fundamental equations: Kinetic theory of granular flow
Particle phase kinetic equation
fi (vi, x, t): number density function of species i
vi: particle velocity
Fi: force acting on each particle (drag, gravity, . . . )
Cij: rate of change of fi due to collisions with species j
Fluid phase equations of motion
∂
βf,i: Drag coefficient of species i.
Characteristic dimensionless parameters
Re = ρf|Uf|L µf
Stokes number - How do particles react to the fluid motion?
Stp = ρpd2
Map = |Up| Θ1/2
Map < 1⇒ Knp = √ π
L
Re = ρf|Uf|L µf
Stp = ρpd2
Map = |Up| Θ1/2
Map < 1⇒ Knp = √ π
L
Re = ρf|Uf|L µf
Stp = ρpd2
Map = |Up| Θ1/2
Map < 1⇒ Knp = √ π
L
Re = ρf|Uf|L µf
Stp = ρpd2
Map = |Up| Θ1/2
Map < 1⇒ Knp = √ π
L
Solution methods for the kinetic equation
∂fi ∂t
∑ j
Cij
Euler-Lagrange method The evolution of each particle position and velocity is tracked individually:
dxα dt
Euler-Euler method The velocity moments of the distribution function are tracked:
M0 i = αp,i =
i fidvi
Solution methods for the kinetic equation
∂fi ∂t
∑ j
Cij
Euler-Lagrange method The evolution of each particle position and velocity is tracked individually:
dxα dt
Euler-Euler method The velocity moments of the distribution function are tracked:
M0 i = αp,i =
i fidvi
Euler-Euler methods: hydrodynamic vs. moment methods
Hydrodynamic models Only low-order moments are tracked: αp,i, Up,i, Θp with Chapman-Enskog expansion to derive constitutive relations Expansion valid for Knp 1
Collision-dominated flows only!
Two-fluid model + constitutive equations for particle properties
Moment methods Moments up to order n are tracked: αp,i, αp,iUp,i, . . . , Mn
Reconstruct velocity distribution function f (v) Use f (v) to close higher-order moments:
Moments spatial fluxes Collision term Force term
Moment transport equations
Euler-Euler methods: hydrodynamic vs. moment methods
Hydrodynamic models Only low-order moments are tracked: αp,i, Up,i, Θp with Chapman-Enskog expansion to derive constitutive relations Expansion valid for Knp 1
Collision-dominated flows only!
Two-fluid model + constitutive equations for particle properties
Moment methods Moments up to order n are tracked: αp,i, αp,iUp,i, . . . , Mn
Reconstruct velocity distribution function f (v) Use f (v) to close higher-order moments:
Moments spatial fluxes Collision term Force term
Moment transport equations
Hydrodynamic models vs. moment methods
Hydrodynamic models Advantages
Unable to predict particle trajectory crossing (Stp limitations)
Difficult to derive physically sound boundary conditions
Moment methods Advantages
Can predict particle trajectory crossing
Easy implementation of Lagrangian BCs
Disadvantages
Computationally inefficient for Kn→ 0
Hydrodynamic models vs. moment methods
Hydrodynamic models Advantages
Unable to predict particle trajectory crossing (Stp limitations)
Difficult to derive physically sound boundary conditions
Moment methods Advantages
Can predict particle trajectory crossing
Easy implementation of Lagrangian BCs
Disadvantages
Computationally inefficient for Kn→ 0
Quadrature-based moment method
The velocity distribution function f (v) is reconstructed using quadrature weights nα and abscissas Uα:
f (v) = β∑ α=1
nαδ(v− Uα)
{M0,M1,M2,M3} ⇔ {nα,Uα : α = 1, 2, . . . , 8}
The method extends to arbitrary order Mn for increased accuracy
MFIX-QMOM implementation
A new module called MFIX-QMOMB has been added to solve the moment transport equations, fully coupled with the existing fluid solver:
∂M0
ijk
Solution process
QMOM solver:
Time integration: 2nd order Runge-Kutta scheme
Space integration: Finite volume method with kinetic flux scheme Time split procedure to account for:
Spatial fluxes Collisions Forces
Solve fluid flow equation
Start QMOM internal loop
due to: Spatial fluxes Collisions Forces
Advance moments of dt QMOM
due to: Spatial fluxes Collisions Forces
t < t old
No
YesNo
Yes
Gas-particle flow in a vertical channel
Channel geometry D = 0.1 m
H = 1.0 m
Re = 1379 < 1500⇒ No turbulence in single-phase flow
Particle phase αp = 0.04
e = ew = 1
H = 1.0 m
e = ew = 1
H = 1.0 m
e = ew = 1
MFIX-QMOM - Two-fluid comparison - St = 0.061 αp - Two-fluid model
MFIX-QMOM
MFIX-QMOM results Steady state profiles (no time average!)
No transient structures A. Passalacqua, R. O. Fox Quadrature-based moment method for polydisperse flows
MFIX-QMOM - Two-fluid comparison - St = 0.061 αp - Two-fluid model
MFIX-QMOM
MFIX-QMOM results Steady state profiles (no time average!)
No transient structures A. Passalacqua, R. O. Fox Quadrature-based moment method for polydisperse flows
MFIX-QMOM - St = 0.061
Mach number Knudsen number Stokes number
MFIX-QMOM results High Mach number flow due to low granular temperature
Knp > 0.1⇒ Hydrodynamic model + partial slip BCs not valid
Stp high enough to give particle trajectory crossing
MFIX-QMOM - St = 0.061
Mach number Knudsen number Stokes number
MFIX-QMOM results High Mach number flow due to low granular temperature
Knp > 0.1⇒ Hydrodynamic model + partial slip BCs not valid
Stp high enough to give particle trajectory crossing
E-L - Two-fluid comparison: St = 0.61
Volume fraction - E-L
Loading movie . . .
Two-fluid model
Loading movie. . .
Mechanism for instability in two-fluid model does not agree with E-L simulations!
eps_d250_densctr.avi
E-L - MFIX-QMOM comparison: St = 0.61
αp - Euler-Lagrange
Loading movie . . .
Loading movie. . .
Mechanism for instability at finite St is identical in E-L and QMOM.
eps_d250_densctr.avi
MFIX-QMOM - Two-fluid comparison: St = 0.61
αp - Two-fluid model
Quantitative difference between two-fluid and QMOM before/after instability
twoFluidShort.mpeg
Conclusions and future work
Conclusions A third-order quadrature-based moment method has been implemented in MFIX
Properly accounting for Kn and St numbers when choosing the model to simulate a gas-particle flow is important
Two-fluid model fails at Kn > 0.1, for high Ma number, and when the St number is high enough for particle trajectory crossing
Future work Implementation of a particle pressure term to enforce the particle packing limit
Further validation against E-L and experimental data for polydisperse cases
Conclusions and future work
Conclusions A third-order quadrature-based moment method has been implemented in MFIX
Properly accounting for Kn and St numbers when choosing the model to simulate a gas-particle flow is important
Two-fluid model fails at Kn > 0.1, for high Ma number, and when the St number is high enough for particle trajectory crossing
Future work Implementation of a particle pressure term to enforce the particle packing limit
Further validation against E-L and experimental data for polydisperse cases
Thanks for your attention!
Overview
Background on kinetic theory of gas-solid flow
MFIX-QMOM implementation

Documents

Development of a quadrature-based moment method for